MATH 2C03, Winter 2019
INTRODUCTION TO DIFFERENTIAL EQUATIONS
To motivate the material, this course starts by examining problems from numerous disciplines where differential equations are used, such as:
population models (exponential growth and decay, logistic equation, Gompertz equation, Allee effect, Von Bertalanffy limited growth model, models involving multiple species such as predator-prey); physics and engineering (free fall, electric circuits, harmonic oscillator, behaviour of cars on a highway, simple pendulum, Newton's law of cooling, diffusion, hanging cable, hydrostatic equilibrium of a star, deflection of a beam, Toricelli's law of draining); biology, chemistry, psychology and medicine (learning curves, blood flow, drug dissolution, lichen growth, ontogenetic growth, basic disease models, chemical reactions, Michaelis-Menten (monod) equation); models in finance and economics (exponential, Solow-Swan, population growth and resource depletion).
We also mention some of the most important partial differential equations, such as heat equation, Laplace's equation, wave equation, Black-Scholes equation, and Schrodinger's equation. The course is fairly technical and involves learning algebraic, geometric, and numeric methods for solving ordinary differential equations, initial and boundary value problems. We start with first-order ODEs, study theorems that guarantee existence and uniqueness of solutions, and then move onto higher order (most often second order) ODEs. We cover some cases, such as constant coefficients and Cauchy-Euler ODEs. Next we cover systems of equations, in particular linear systems with constant coefficients, and use linear algebra to solve them (eigenvalues and eigenvectors and matrix exponentials). We briefly visit systems of non-linear ODEs and introduce phase plane diagrams. As an application of techniques of solving ODEs we discuss separation of variables in PDEs, in particular the way it is applied to first order PDEs as well as to the heat equation and Laplace's equation.
First order ordinary differential equations and higher order linear ordinary differential equations including Laplace transforms and series solutions.
Three lectures, one tutorial; one term
Prerequisite(s): One of MATH 1AA3, 1LT3, 1NN3, 1XX3, 1ZB3, ARTSSCI 1D06 A/B, ISCI 1A24 A/B; and one of MATH 1B03, 1ZC3
Antirequisite(s): ENGINEER 2Z03, MATH 2M03, 2M06, 2P04, 2Z03
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